The relevant part of the next heat kernel expansion coefficient is (see also [1]}, [2]})
\(\begin{array}{l}\tilde{a}_3=\frac{1}{3!}\Big [\frac{2}{15}\tilde{\nabla }_{\mu }\Omega _{\nu \rho }\tilde{\nabla }^{\mu }\Omega ^{\nu \rho }+\frac{1}{30}(\tilde{\nabla }^{\nu }\Omega _{\nu \mu })^2+\frac{1}{5}\Omega ^{\mu \nu }\tilde{\nabla }^2\Omega _{\mu \nu }+\frac{1}{10}\tilde{R}_{\alpha \beta \mu \nu }\Omega ^{\alpha \beta }\Omega ^{\mu \nu }+\frac{1}{15}\tilde{R}^{\mu \nu }\Omega ^2_{\mu \nu }-\\-\frac{1}{10}(\tilde{\nabla }^2)^2E-\tilde{\nabla }_{\mu }V\tilde{\nabla }^{\mu }E-\frac{1}{15}\tilde{R}_{\mu \nu }\tilde{\nabla }^{\mu }\tilde{\nabla }^{\nu }E-\frac{1}{5}\tilde{\nabla }^{\mu }\tilde{R}\tilde{\nabla }_{\mu }E+\frac{1}{2}\tilde{\nabla }_{\mu }E\tilde{\nabla }^{\mu }E \Big ],\end{array}\) 
